Theorem seqeq1 10559

Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Assertion

Ref	Expression
seqeq1	|- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )

Proof of Theorem seqeq1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . . . 6 |- ( M = N -> M = N )
2		fveq2 5561	. . . . . 6 |- ( M = N -> ( F ` M ) = ( F ` N ) )
3	1, 2	opeq12d 3817	. . . . 5 |- ( M = N -> <. M , ( F ` M ) >. = <. N , ( F ` N ) >. )
4		freceq2 6460	. . . . 5 |- ( <. M , ( F ` M ) >. = <. N , ( F ` N ) >. -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
5	3, 4	syl 14	. . . 4 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
6		fveq2 5561	. . . . . 6 |- ( M = N -> ( ZZ>= ` M ) = ( ZZ>= ` N ) )
7		eqid 2196	. . . . . 6 |- _V = _V
8		mpoeq12 5986	. . . . . 6 |- ( ( ( ZZ>= ` M ) = ( ZZ>= ` N ) /\ _V = _V ) -> ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) )
9	6, 7, 8	sylancl 413	. . . . 5 |- ( M = N -> ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) )
10		freceq1 6459	. . . . 5 |- ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
11	9, 10	syl 14	. . . 4 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
12	5, 11	eqtrd 2229	. . 3 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
13	12	rneqd 4896	. 2 |- ( M = N -> ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = ran frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
14		df-seqfrec 10557	. 2 |- seq M ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
15		df-seqfrec 10557	. 2 |- seq N ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. )
16	13, 14, 15	3eqtr4g 2254	1 |- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )

Syntax hints:   -> wi 4   = wceq 1364   _Vcvv 2763   <.cop 3626   ran crn 4665   ` cfv 5259  (class class class)co 5925   e. cmpo 5927  freccfrec 6457   1c1 7897   + caddc 7899   ZZ>=cuz 9618   seqcseq 10556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-cnv 4672  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fv 5267  df-oprab 5929  df-mpo 5930  df-recs 6372  df-frec 6458  df-seqfrec 10557

This theorem is referenced by:  seqeq1d  10562  seq3f1olemqsum  10622  seqf1oglem2  10629  seq3id  10634  seq3z  10637  iserex  11521  summodclem2  11564  summodc  11565  zsumdc  11566  isumsplit  11673  ntrivcvgap  11730  ntrivcvgap0  11731  prodmodclem2  11759  prodmodc  11760  zproddc  11761  fprodntrivap  11766  ege2le3  11853  gsumfzval  13093  gsumval2  13099